The two nations entered play tied at 1-1 after splitting Friday's singles matches, but Dodig and Karlovic gave Croatia the lead in the best-of-five tie with a 6-4, 6-4, 3-6, 6-3 victory over the host nation's pairing of Ito and Sugita.
The Croats converted on just 3-of-14 break chances, but reeled off 24 aces en route to the win.
Croatia will now look to take the first meeting between the two nations with Sunday's reverse singles, which currently call for Japan's Kei Nishikori to battle Dodig and Karlovic to take on Go Soeda.
Croatia captured its lone Davis Cup title in 2005, while Japan has only ever reached one final, and that resulted in a loss against the 32-time champion United States way back in 1921.
The Croatia-Japan winner will meet the Argentina-Germany victor in the quarterfinals in April.